This note is devoted to a short overview of some results on derived categories of coherent sheaves concerning smooth categorical compactifications and the formal punctured neighborhoods of infinity.
In Section 1, we discuss the conjecture of Bondal and Orlov about the categorical properties of the resolution of singularities of an algebraic variety with rational singularities (Conjecture 1.1). This conjecture states that the derived pushforward functor on the derived categories of coherent sheaves is a quotient functor (that is, a localization). The conjecture is difficult and still open in general. It turns out that it is possible (Theorem 1.2) to prove a version of such statement for an arbitrary separated scheme of finite type over a field of characteristic zero (the reader may safely assume that we are dealing with quasi-projective schemes). The methods make it possible to prove Conjecture 1.1 for a cone over a projective embedding of a smooth Fano variety (that is, a smooth projective variety with an ample anti-canonical line bundle).
In Section 2, we consider DG categorical smooth compactifications. Here DG stands for “differential-graded”. This is a straightforward generalization of the usual algebro-geometric smooth compactification. The following natural question was formulated by B. Toën (Question 2.3 below): is it true that any smooth DG category “of finite type” admits a smooth categorical compactification? The question was considered to be difficult, but most experts expected that the answer should be “yes”. However, in  we gave a negative answer, obtained by disproving a closely related conjecture of Kontsevich (Conjecture 2.5 below) on the generalized version of the degeneration of the Hodge-to-de-Rham spectral sequence. We also obtained a dual version of these results, in which smooth DG categories are replaced by proper DG categories, and a smooth compactification is replaced by a categorical resolution of singularities.
In Section 3 we outline a certain construction called a “categorical formal punctured neighborhood of infinity”. For a smooth algebraic variety this is obtained as follows: take some smooth compactification consider the formal completion at the infinity locus and then take the corresponding punctured formal scheme. The resulting object (considered for example as an adic space) is independent of the compactification, as is the category of perfect complexes on it. In  we give a purely categorical construction of which generalizes to arbitrary smooth DG algebras and DG categories. A curious special case is the algebra of rational functions on a smooth projective curve. There, our construction gives exactly the ring of adeles.
1 Rational singularities and a conjecture of Bondal and Orlov
Let be an algebraic variety over a field of characteristic zero. Recall that has rational singularities if for some (and then any) resolution of singularities we have Equivalently, the pullback functor is fully faithful. The following conjecture is still open.
Conjecture 1.1 (). With the above notation, the functor is a localization. That is, the induced functor is an equivalence.
The following result is a version of such statement which holds in a much more general framework.
Let be a separated scheme of finite type over a field of characteristic zero. Then there exist a smooth projective variety and a functor such that the induced functor is an equivalence. Moreover, the triangulated category is generated by a single object.
This theorem in particular confirms a conjecture of Kontsevich on the homotopy finiteness of the DG category The proof is based on a certain construction of a categorical resolution of singularities, due to Kuznetsov and Lunts .
Let be a cone over a smooth Fano variety in Let be the resolution given by the blow-up of the origin point. Then the induced functor is an equivalence.
2 Categorical smooth compactifications
Theorem 1.2 deals with a special case of a categorical smooth compactification. We first recall some basic definitions.
Definition 2.1 ().
A small DG category over is smooth if the diagonal -bimodule is perfect.
is called proper if for the complex is perfect over
In particular, we have the notions of smoothness and properness for DG algebras (a DG algebra can be considered as a DG category with a single object). When is a separated scheme of finite type over a field then is smooth (resp. proper) if and only if the DG category is smooth (resp. proper) ([11, Proposition 3.30], [10, Proposition 3.13]). Hence, these basic geometric properties of are reflected by the DG category
We recall the following definition.
Definition 2.2. For a pre-triangulated DG category a categorical smooth compactification is a DG functor such that:
is a smooth and proper pre-triangulated DG category;
the induced functor is fully faithful;
every object is a direct summand of some .
The basic geometric example of a categorical smooth compactification is given by the usual one. Namely, let be a smooth algebraic variety over and let be an open embedding, where is smooth and proper. Then the restriction functor is a categorical smooth compactification.
Theorem 1.2 provides a categorical smooth compactification of the DG categories of the form where is a separated scheme of finite type over a field of characteristic zero.
There is a notion of a homotopically finitely presented (hfp) DG category which should be thought of as a smooth DG category "of finite type" (we refer to  for the precise definition). The following general question was formulated by Bertrand Toën.
Question 2.3 (Toën). Is it true that any homotopically finitely presented DG category over a field of characteristic zero has a smooth compactification?
The question is difficult, but the general consensus was that the answer should be “yes”. However, in  the author gave a negative answer to this question. Here we explain the rough idea of the results of .
It turns out that Question 2.3 is closely related with the non-commutative (categorical) Hodge-to-de Rham degeneration. Recall that the classical Hodge theory implies (via GAGA) the following algebraic statement: for any smooth algebraic variety over a field of characteristic zero the spectral sequence
Let be a smooth and proper DG algebra over a field of characteristic zero. Then the Hochschild-to-cyclic spectral sequence degenerates, so that we have an isomorphism .
In the special case when for a smooth and proper variety Theorem 2.4 gives exactly the usual (commutative) Hodge-to-de Rham degeneration.
The following two conjectures were formulated by Kontsevich for smooth and for proper DG algebras.
Conjecture 2.5 (Kontsevich). Let be a smooth DG algebra over a field of characteristic zero. Then the composition
vanishes on the class of the diagonal bimodule.
Here denotes the boundary map, see [2, Section 3].
Conjecture 2.6 (Kontsevich). Let be a proper DG algebra over a field of characteristic zero. Then the composition map
Here denotes the boundary map, see [9, Section 2.2].
A dual argument implies that Conjecture 2.6 holds for proper DG categories which can be fully faithfully embedded into a smooth and proper DG category (such an embedding is called a categorical resolution in the terminology of Kuznetsov and Lunts ).
However, in  we disproved both conjectures.
There exists a homotopically finitely presented DG algebra for which Conjecture 2.5 does not hold. In particular, gives a negative answer to Question 2.3: the DG category does not have a smooth categorical compactification.
There exists a proper DG algebra for which Conjecture 2.6 does not hold. In particular, the category does not have a categorical resolution of singularities.
The DG algebra from part 2 is quasi-isomorphic to a certain explicit -dimensional -algebra for which the supertrace of on the second argument is non-zero.
3 Categorical formal punctured neighborhood of infinity
Another subject related to the notion of a smooth categorical compactification is that of a formal punctured neighborhood of infinity. Suppose that we have a usual smooth compactification of a smooth algebraic variety . Then one can take the formal neighborhood and then “remove” . The resulting object (the so-called generic fiber, considered as an adic space) does not depend on the choice of the compactification Let us set The corresponding category of perfect complexes also does not depend on and it is therefore an invariant of .
The natural question arises: can we describe the category purely in terms of ? This question is partially motivated by mirror symmetry since an analogue of exists in symplectic geometry in the framework of Fukaya categories. It turns out that the purely categorical construction is possible, and it was described by the author in . Here we give an outline.
First, we describe a “non-derived” version of the construction. Let be an associative algebra over a field Then one can describe the algebra as follows.
Here is the algebra of -linear endomorphisms of (as a vector space) and is the two-sided ideal of operators of finite rank. The commutator is the additive one (the Lie algebra bracket) and is the operator of right multiplication by
Example 3.1. It is a pleasant exercise to check that for we have A similar computation shows that .
Example 3.2. A less trivial example is the following: let be a smooth projective connected curve over Then we have where is the ring of adeles on . Recall that is the subring of the product of complete local fields, consisting of elements such that for all but finitely many .
Now let be a smooth DG algebra. The DG algebra is defined by the formula
Here denotes the Hochschild cochain complex. The product on comes from the product on .
To describe the DG algebra more conceptually, we recall the following notion.
Let be a field. The Calkin (DG) category is defined as the quotient More explicitly, the objects of the DG category are complexes of -vector spaces, and the morphisms are given by .
More generally, for a DG algebra the Calkin category is defined as the quotient .
We can consider (and any other right -module) as an object of – suitably defined category of representations of in . Note that
The DG category of topological perfect complexes over is defined as follows.
Definition 3.4. For a smooth DG algebra we define
Here the embedding comes from the assumption that is smooth. The functor is given by the tensor product: for .
Let be a smooth algebraic variety over a field and assume that has a smooth compactification. Let be a DG algebra such that Then we have an equivalence such that the following diagram commutes:
Remark 3.6. It is possible to obtain an extended version of Theorem 3.5 where the category is replaced by the category of nuclear modules in the sense of Clausen and Scholze [12, Definition 13.10]). This is more involved (and unpublished), and we will not cover this in the present note.
Remark 3.7. The construction of the DG algebra and the DG category is very much in the spirit of Tate’s paper on residues of differential on curves .
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Cite this article
Alexander I. Efimov, Categorical smooth compactifications and neighborhoods of infinity. Eur. Math. Soc. Mag. 120 (2021), pp. 4–7DOI 10.4171/MAG/34